This is a question frequently asked during technical interviews. I will try to answer it through this short article giving examples and the logic behind them. Here are the main points I deal with here:

*Reminder: Gamma and Theta**Black Scholes vs Real World**Vertical Spread**Conclusion – Smile skew*

*Gamma*

*Gamma* is defined as the second derivative of the option price with respect to the underlying. *Gamma* for a vanilla call and put is given by:

with the normal density function. A long position on an option makes you *Gamma* positive. *Gamma* tells us how our delta will change for a 1$ move in the underlying.

*Theta (silent killer)*

*Theta* is defined as the first derivatives of the option price with respect to the time. *Theta* for a vanilla call is given by:

The formula for the put option can be derived with the call-put parity. *Theta* is the decay of the extrinsic value (time value) of an option, all else equal. One has a positive *Theta* when selling an option and a negative *Theta* when buying it.

*Black & Scholes*

Within the B&S world, volatility is flat. Options on the same underlying will have the same volatility whatever the strike and the maturity chosen. In this case, it’s impossible to find a combination leading to a positive gamma and theta (it can be proven from the Black&Scholes PDE that Gamma and Theta are opposite signs).

*Reality*

In reality, we all know volatility for all options isn’t flat. We talk about implied volatility and it depends on the strike and the maturity of your options. It’s deduced from the market price of each option and therefore by supply and demand. This screenshot shows a set of implied volatilities for different strikes and maturities:

Notice the pattern here: OTM (Out The Money) puts have higher implied volatility than OTM calls (spot is 24.74€).

The smile changes everything for our research. Actually, Greeks are sensitive to volatility. Let’s illustrate this with a little example I made on Excel:

- Call S=45 K=50 T=1Y vs

We can see here that every greeks changed. Now let’s focus on *Theta*: with a higher volatility and all else equal, we finally get a higher Theta in absolute value.

Now let’s plot the gamma for different spot value:

We can notice that *Gamma *is max for ATM options and decreases for ITM and OTM options.

Here we are, we have everything we need to make a *Theta* and *Gamma* positive combination! Being *Theta* and *Gamma* positive is possible when shorting the high volatility (OTM option) and going long the low volatility (near ATM).

*Bull Call Spread*

A bull call spread is a combination in which you go long a call and short a call with . Let’s see what happen when inserting a big smile into the call spread pricer:

We did it, we are *Gamma* and *Theta* positive.

*Insight*: the volatility gap between the two calls is so big that the OTM call *Theta *is higher than the ATM call *Theta.*

*Bear Put Spread*

Naturally, we can also find a Bear Put Spread that leads to the same kond of result. All we need is a smile, sell the OTM put and buy ATM put:

We are *Gamma* and *Theta* positive, the logic behind is exactly the same.

*Horizontal Spread*

I tried to find another trick playing with a calendar spread but I did not find a plausible combination that leads to a positive G*amma* and *Theta. *In fact, it works in inserting a really dropping implied volatility for the longer maturity but it joins the same logic as we saw just before.

Consider 2 options near the money with same strike, one long-term and one short-term. You sell the short-term and buy the long-term. The short-term option has both a higher *Theta* and *Gamma *than the long-term and then it leads to a positive *Theta* and negative *Gamma*. Indeed, for ATM options, *Theta *and *Gamma* increase when expiration is coming.

However, if you include a lower volatility (short-term uncertainty about a company?) for the long-term option its *Gamma *will increase and may exceed the *Gamma* of the short-term option. Example :

*Conclusion*

*Theta* and *Gamma* positive combinations exist in the real world. However, it’s not that easy to find for some reasons:

**First**, we need a large implied volatility gap between our two options:

- In the vertical spread, so that the OTM
*Theta*is higher in absolute value than the ATM*Theta*while not going too far away out the money. - In the horizontal spread, so that short-term option has less
*Gamma*than the long-term option.

**Second**, in the equity derivatives market, OTM calls have lower implied volatility OTM puts. We usually find skew in the smile, like this one:

Reason for this skew is the fear of the downside. Investors are long stock in general and perceive the risk of a crash much greater than the risk of the soaring price. Thus they buy OTM puts as a protection.

In this market, it will be easier to find a Put Spread leading to a positive *Gamma* and *Theta.*

In commodities, we see the converse. OTM alls have higher implied volatilities than OTM puts. Companies run their business with commodities, their margin disappears if prices go up, they fear the upside. Then they buy OTM calls a protection.

In this market, it will be easier to find a call spread leading to a positive *Gamma* and *Theta.*

*Thank you!*

*Thank you!*

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